Digital holography can highly accurately measure phase information which cannot be directly detected by a light detector, and as such plays a very important role in the field of three-dimensional measurement of object shapes, biological samples, and the like. In digital holography, a hologram (interference fringes) made from an object beam from an inspection object and a reference beam which can interfere with the object beam is acquired in the form of a digital image with use of an imaging device, and, from the distribution of the interference fringes, the intensity distribution and the phase distribution (complex amplitude distribution) of the object beam are computed. Digital holography in which the propagation angle of the object beam and the propagation angle of the reference beam are different at the time when a hologram is made as illustrated in FIG. 1 is called “off-axis digital holography.” A digital holography in which the propagation angle of the object beam and the propagation angle of the reference beam are identical to each other is called “on-axis digital holography.”
The following describes the principles and problems of spatial filtering and phase-shifting interferometry, which are typical methods for measuring a complex amplitude distribution in digital holography.
(Spatial Filtering)
In off-axis digital holography, a complex amplitude of an object beam can be computed from one hologram (Non-PTL 1). When a reference beam which is a plane wave having a uniform intensity distribution and a uniform phase interferes with an object beam while being shifted by angle θ with respect to the object beam as illustrated in FIG. 1, hologram H made from object beam O (Expression (1)) and reference beam R (Expression (2)) is expressed by Expression (3).[Expression 1]O=A0exp(iφ)  (1)R=Arexp(iψ+iθ)  (2)H=|O|2+|R|2+O*R+OR*  (3)
where Ao and Ar are amplitudes of object beam O and reference beam R, respectively, and φ and ψ are phases of object beam O and reference beam R, respectively. In Expression (3), the first term and second term on right side represent the zero-order beam (direct current component), the third term represents the + primary beam (conjugate image), and the fourth term represents the − primary beam (real image).
The − primary beam, an object beam component, can be extracted from the above-mentioned beams by applying Fourier transform to Expression (3) by virtual reference beam RD (Expression (4)) which is called a digital reference beam and simulated in a calculator, and by performing spatial filtering in accordance with Nyquist aperture W around the center of the frequency of the real image to perform inverse Fourier transform (Expression (5)), as illustrated in FIG. 2.[Expression 2]RD≡1/R*  (4)O=IFT{FT[H1RD]×W}  (5)
where FT and IFT are Fourier transform and inverse Fourier transform, respectively.
In order to effectively extract the − primary beam in the frequency space, it is necessary to sufficiently increase angle θ. However, there is a problem that the resolution of the object beam component significantly decreases as angle θ increases. In addition, there is a limitation that the real image cannot be properly reproduced when angle θ is greater than maximum incident angle θmax (Expression (6)) defined by the Nyquist frequency.
                    [                  Expression          ⁢                                          ⁢          3                ]                                                                      θ          max                =                  arc          ⁢                                          ⁢                      sin            ⁡                          (                              λ                                  2                  ⁢                                                                          ⁢                  Δ                  ⁢                                                                          ⁢                  x                                            )                                                          (        6        )            
where λ represents the wavelength of an object beam and a reference beam, and Δx represents the pixel width of the imaging device.
Expression (6) means that interval Λ of the interference fringes (Expression (7)) is excessively narrowed in comparison with pixel width Δx of the imaging device when θ is greater than θmax (θ>θmax). This suggests that the sampling theorem cannot be satisfied in the recorded hologram. For example, when wavelength λ is 0.532 μm and the pixel width Δx is 3.75 μm, maximum incident angle θmax is 4.07°.
                    [                  Expression          ⁢                                          ⁢          4                ]                                                            Λ        =                  λ                      2            ⁢                                                  ⁢            sin            ⁢                                                  ⁢            θ                                              (        7        )            
As described above, the greater angle θ between the object beam and the reference beam, the − primary beam can be more effectively extracted. When a record-reproduce simulation of a digital holography is performed with θ being set at 4.00 degrees which is close to maximum incident angle θmax in the above-mentioned example, a striped pattern is undesirably formed in the reproduced image, since the zero-order beam and the − primary beam are not sufficiently separated as illustrated in FIG. 3. In order to separate the zero-order beam and the − primary beam from each other, it is necessary to calculate the diffraction propagation of the reproducing image by Fresnel diffraction integral and the like to compute the complex amplitude of the distance where the zero-order beam and the − primary beam are not spatially superimposed in a numerical analysis manner. This, however, requires to adjust the distance between the object and the imaging device such that the zero-order beam and the − primary beam can be spatially separated from each other. In such a configuration, the size of the optical system is large, and high-frequency components do not incident on the imaging device, and therefore it is difficult to achieve high resolution.
Under such circumstances, a method has been proposed in which a − primary beam is efficiently extracted even when angle θ is small (Non-PTL 2). In the method disclosed in Non-PTL 2, two reference beams with different phases are used to make two holograms, and the difference thereof is obtained to eliminate the zero-order beam component, whereby a real image is effectively computed. Two holograms H1 and H2 are expressed by Expression (8) and Expression (9), respectively, with the phase difference between the holograms being ψ.[Expression 5]H1=|O1|2+|R12+O1*R1+O1R1*  (8)H2=|O2|2+|R2|2+O2*R2+O2R2*  (9)
where object beams O1 and O2 making two holograms H1 and H2 are expressed by Expression (10) and Expression (11), respectively. In addition, reference beams R1 and R2 making two holograms H1 and H2 are expressed by Expression (12) and Expression (13), respectively.[Expression 6]O1=Ao1exp(iφ1)  (10)O2=Ao2exp(iφ2)  (11)R1=Ar1exp(iψ1+iθ)  (12)R2=Ar2exp(iψ2+iθ)  (13)
In the method disclosed in Non-PTL 2, Ao1, Ao2, Ar1, Ar2, φ1 and φ2 are assumed as in Expression (14) to Expression (16), and the object beam and the reference beam are assumed as in Expression (17) and Expression (18), respectively, whereby Expression (8) and Expression (9) are rewritten as Expression (19) and Expression (20).[Expression 7]Ao≡Ao1=Ao2  (14)Ar≡Ar1=Ar2  (15)φ≡φ1=φ2  (16)O≡Aoexp(iφ)  (17)R≡Arexp(iθ)  (18)H1=|O|2+|R|2+O*Rexp(iψ1)+OR*exp(−iψ1)  (19)H2=|O|2+O*Rexp(iψ2)+OR*exp(−iψ2)  (20)
The difference between Expression (19) and Expression (20) is expressed by Expression (21), in which the zero-order beam is eliminated and only the ± primary beams remain. With use of this relationship, the object beam component can be computed as expressed by Expression (22) similarly to Expression (5).
                    [                  Expression          ⁢                                          ⁢          8                ]                                                                                  H            1                    -                      H            2                          =                                            O              *                        ⁢                          R              ⁡                              [                                                      exp                    ⁡                                          (                                              ⅈ                        ⁢                                                                                                  ⁢                                                  ψ                          1                                                                    )                                                        -                                      exp                    ⁡                                          (                                              ⅈ                        ⁢                                                                                                  ⁢                                                  ψ                          2                                                                    )                                                                      ]                                              +                                    OR              *                        ⁡                          [                                                exp                  ⁡                                      (                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                                              ψ                        1                                                              )                                                  -                                  exp                  ⁡                                      (                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                                              ψ                        2                                                              )                                                              ]                                                          (        21        )                                                          ⁢                  O          =                                    IFT              ⁢                              {                                                      FT                    ⁡                                          [                                                                        (                                                                                    H                              1                                                        -                                                          H                              2                                                                                )                                                ⁢                                                  R                          D                                                                    ]                                                        ×                  W                                }                                                                    exp                ⁡                                  (                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          ψ                      1                                                        )                                            -                              exp                ⁡                                  (                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          ψ                      2                                                        )                                                                                        (        22        )            
When the method disclosed in Non-PTL 2 is applied, the zero-order beam component, which cannot be completely eliminated by the conventional off-axis digital holography (see FIG. 2), can be sufficiently removed as illustrated in FIG. 4 and FIG. 5. However, with the method disclosed in Non-PTL 2, since an image is reproduced after spatial filtering is performed, the image is undesirably blurred (see FIG. 5). In addition, since Fourier transform is required to be performed twice before the complex amplitude is computed, the calculation cost is high, and discretization errors and calculation errors of numerical calculations are undesirably accumulated. As a result, it is difficult to measure complex amplitude with high accuracy. Further, when Fourier transform is performed, it is necessary to expand the analysis area at least fourfold in order to enhance the calculation accuracy, and a process (zero padding) for appending zero to the expanded part is required to be performed. When the number of pixels of the imaging device is represented by N2, the calculation quantity is 4N2 log (2N) even when only the part subject to Fourier transform is taken into consideration. Accordingly, the computer is required to have a sufficient memory region and a sufficient arithmetic speed. As such, in the off-axis digital holography, it is difficult to measure phase information with high accuracy.
(Phase-Shifting Interferometry)
On the other hand, on-axis digital holography can maximize the resolution of an imaging device. For this reason, in recent years, on-axis digital holography is most broadly studied and developed as a method having a potential to detect a complex amplitude of an object beam with high definition and high accuracy. In on-axis digital holography, three optical waves, a zero-order beam and ± primary beams, propagate at the same angle, and therefore the waves cannot be effectively separated by a space filter. Therefore, in on-axis digital holography, a phase measurement method called phase-shifting interferometry is used.
Digital holography utilizing phase-shifting interferometry is called phase-shift digital holography. In phase-shift digital holography, multiple holograms in which phases of reference beams are different from each other are recorded by an imaging device, and complex amplitude information of an object beam is computed from the holograms. The type of phase-shift digital holography is classified by the method of changing the phase of a reference beam, number of required holograms, and the like. For example, phase-shift digital holography is roughly categorized by the method of changing the phase of a reference beam into sequential type (sequential phase-shift digital holography) in which the phase is changed in a time-dependent manner, and parallel type (parallel phase-shift digital holography) in which the phase is spatially changed. In addition, phase-shift digital holography is roughly categorized by the required number of holograms into a 4-step method using four holograms, a 3-step method using three holograms, a 2-step method using two holograms, and the like. It is to be noted that, in phase-shift digital holography, at least two holograms are required.
Sequential phase-shifting interferometry is a scheme in which the phase of a reference beam is sequentially changed with use of a piezoelectric element and the like to sequentially acquire multiple holograms. FIG. 6 illustrates a configuration of a sequential phase shift interferometer used in sequential phase-shifting interferometry. FIG. 7 shows procedures of sequential phase-shifting interferometry (4-step method), and FIG. 8 shows procedures of sequential phase-shifting interferometry (2-step method). In FIGS. 7 and 8, O1 to O4, and O represent object beams, R1 to R4 represent reference beams, and H1 to H4 represent holograms. As illustrated in FIGS. 7 and 8, in the sequential scheme, phases of object beams are computed from holograms recorded at different times, and therefore a large measurement error may possibly be caused when phase information of an object beam from an object that changes with time (for example, microbe) is measured.
Parallel phase-shifting interferometry is a scheme in which a reference beam is spatially split, and the phase is changed for each of the split reference beams with use of a phase shift array device and the like to simultaneously acquire multiple holograms. FIG. 9 illustrates a configuration of a phase-shifting interferometer used in parallel phase-shifting interferometry. FIG. 10 illustrates an operation of a phase shift array device in a phase-shifting interferometer. FIG. 11 illustrates division of a hologram and interpolation of pixels in parallel phase-shifting interferometry. FIG. 12 illustrates procedures of in parallel phase-shifting interferometry (2-step method). In FIGS. 12, O1 and O represent object beams, R0 to R2 represent reference beams, and H, H1, and H2 represent holograms. Parallel phase-shifting interferometry can measure phase information in a short time, but at the same time has a disadvantage that a measurement error in association with the interpolation is inevitably caused (see FIG. 11).
As described above, in the phase-shifting interferometry, at least two holograms are required. A method of reproducing an object beam from two holograms was first proposed by Gabor et al. in 1966 (Non-PTL 3). The method proposed by Gabor el at., however, has the problem of complicated optical system. For this reason, today, methods have been broadly used which can be achieved with a simple optical system, with which an object beam is reproduced from three (Non-PTL 4) or four (Non-PTL 5) holograms. In particular, phase-shifting interferometry of a sequential scheme using four holograms illustrated in FIG. 7 has been most frequently used because of its simple calculation formula. The 4-step method can be achieved with a simple optical system; however, the 4-step method requires a large number of holograms and lacks high-speed performance, and therefore the temporal change of an object beam cannot be handled. Likewise, parallel phase-shifting interferometry using four holograms requires a large number of holograms, and thus has the problem of occurrence of large interpolation errors. Further, in the 4-step method, only the proportional relationship of the amplitude can be evaluated, and the proportional constant cannot be computed, and therefore, the amplitude value cannot be determined.
Under such circumstances, sequential phase-shifting interferometry (2-step method) has been proposed in which an object beam can be reproduced from two holograms with use of a simple optical system illustrated in FIG. 8 (Non-PTL 6). In this method, from Expression (19) and Expression (20) in which θ=0, ψ1=0, and ψ2=π/2, the complex amplitude of object beam O is obtained as follows (Expression (23) to Expression (26)).
                                              ⁢                  [                      Expression            ⁢                                                  ⁢            9                    ]                                                                                              ⁢                              H            1                    =                                                                                      A                  o                                                            2                        +                                                                            A                  r                                                            2                        +                          2              ⁢                                                          ⁢                              A                0                            ⁢                              A                r                            ⁢              cos              ⁢                                                          ⁢              ϕ                                                          (        23        )                                                          ⁢                              H            2                    =                                                                                      A                  o                                                            2                        +                                                                            A                  r                                                            2                        +                          2              ⁢                                                          ⁢                              A                0                            ⁢                              A                r                            ⁢              sin              ⁢                                                          ⁢              ϕ                                                          (        24        )                                                                                                                ⁢                              O                =                                ⁢                                                      A                    o                                    ⁢                                      exp                    ⁡                                          (                                              ⅈ                        ⁢                                                                                                  ⁢                        ϕ                                            )                                                                                                                                              =                            ⁢                                                                                          H                      1                                        -                    I                                                        2                    ⁢                                                                                  ⁢                                          A                      r                                                                      +                                  ⅈ                  ⁢                                                                                    H                        2                                            -                      I                                                              2                      ⁢                                                                                          ⁢                                              A                        r                                                                                                                                                    (        25        )                                          2          ⁢                                          ⁢          I                =                              H            1                    +                      H            2                    +                      2            ⁢                                                  ⁢                          A              r              2                                -                                    [                                                                    (                                                                  H                        1                                            +                                              H                        2                                            +                                              2                        ⁢                                                                                                  ⁢                                                  A                          r                          2                                                                                      )                                    2                                -                                  2                  ⁢                                      (                                                                  H                        1                        2                                            +                                              H                        2                        2                                            +                                              4                        ⁢                                                                                                  ⁢                                                  A                          r                          4                                                                                      )                                                              ]                                      1              /              2                                                          (        26        )            
While the phase of the reference beam may have any values, ψ1=0 and ψ2=π/2 which are widely used are assumed in this case. As illustrated in FIG. 8, in this method, intensity distribution |R1|2 of the reference beam is required to be preliminary recorded before the measurement. However, after intensity distribution |R1|2 of the reference beam is once recorded and stored, the phase measurement can be continuously performed unless its distribution is changed. In addition, while a reference beam having a uniform intensity distribution is required to be prepared in the 4-step method, the intensity distribution of a reference beam is not required to be uniform in the 2-step method according to Non-PTL 6 as long as the intensity of the reference beam is more than double the intensity of the object beam. Further, the 2-step method according to Non-PTL 6 can compute the value of amplitude, and therefore is highly usable.
The 2-step method according to Non-PTL 6 is a breakthrough method in which the required number of holograms can be reduced from four to two with use of a simple optical system. In the 2-step method according to Non-PTL 6, the calculation quantity is N2, and therefore calculation cost can be considerably reduced in comparison with the above-mentioned method that requires Fourier transform. However, since it is necessary to image at least two holograms having different phases, the frame rate of the imaging device decreases to one-half or less in the case of the sequential scheme. Thus, the 2-step method according to Non-PTL 6 has a problem that a large measurement error is caused when the phase information of an object which changes with time is measured.
In addition, when the 2-step method according to Non-PTL 6 is applied in the parallel scheme, it is necessary to modulate the phase of a reference beam in a unit of at least two pixels with use of a special phase shift array device as illustrated in FIG. 10. Thus, disadvantageously, the resolution of the imaging device is halved, and interpolation errors for interpolating the resolution are inevitably caused as illustrated in FIG. 11 (Non-PTL 7). Here, the “interpolation error” does not refers to simple reduction in resolution, but refers to increase in measurement error caused by the phase measurement which is performed on the basis of the interpolated information. The interpolation error cannot be prevented from occurring by increasing the resolution of the imaging device. In addition, the phase shift array device causes unnecessary diffraction phenomenon, which is a cause of noise. Further, in the 2-step method according to Non-PTL 6, the spatial division for acquiring a hologram is not assumed unlike parallel phase-shifting interferometry, and therefore only one of intensity distributions (|R1|2 and |R2|2) of two reference beams R1 and R2 is allowed to be taken into consideration as illustrated in FIG. 12. In general, there is a difference between these intensity distributions, and thus measurement errors are caused.
A diversity scheme (diversity phase-shifting interferometry) has been proposed (PTL 1) as illustrated in FIG. 13 and FIG. 14, as a method for solving both of the problems of the above-mentioned sequential scheme and parallel scheme. In the diversity phase-shifting interferometry, an object beam and a reference beam are not spatially split, but multiple holograms are simultaneously recorded by generating copies of an object beam and a reference beam with use of a beam splitter. Thus, a complex amplitude distribution of an object beam can be measured with high speed and high accuracy without causing reduction in resolution and interpolation errors. The interferometer illustrated in FIG. 13 is a 4-channel holographic diversity interferometer for use in the case where the 4-step method is applied (FIG. 8 of PTL 1). In FIG. 13, “BS” is a beam splitter, “PBS” is a polarization beam splitter, “HWP” is a ½ wavelength plate, and “QWP” is a ¼ wavelength plate. In this interferometer, the required number of imaging devices can be reduced by designing the optical arrangement with use of a mirror and the like; however, the optical system is complicated and the size is increased since it is necessary to ensure four image regions. Under such circumstances, as illustrated in FIG. 15 and FIG. 16, a two-channel holographic diversity interferometer has been proposed (FIG. 15 of PTL 1) in which the number of holograms to be simultaneously recorded is reduced by adopting the 2-step method.
Holographic diversity interferometry (2-step method) can considerably simplify the optical system to downsize the apparatus. Holographic diversity interferometry (2-step method), however, can take the intensity distribution of only one of two reference beams into consideration, as with parallel phase-shifting interferometry. For this reason, when there is a difference between the intensity distributions of two reference beams, measurement error is caused. In an actual system, a difference in intensity distributions of an object beam and a reference beam may possibly be caused due to difference in light receiving sensitivity between imaging devices, the roughness of the surface shape of the optical device and the like, thus reducing the accuracy of measurement.
In all of the above-mentioned sequential scheme, parallel scheme and diversity scheme, the measurement accuracy is largely dependent on the difference in intensity distributions of an object beam and a reference beam between holograms in the phase-shifting interferometry. Table 1 shows characteristics of the above-mentioned schemes. FIGS. 17A to 17F show measurement errors which may possibly be caused in the schemes. The measurement errors shown in FIGS. 17A to 17F were analytically computed on the basis of Expression (23) to Expression (26). The “Intensity ratio of reference beams” in FIGS. 17A and 17B are defined by (intensity of object beam of hologram H2)/(intensity of object beam of hologram H1). The “Intensity ratio of reference beams” in FIGS. 17E and 17F is defined by (intensity of reference beam of hologram H2)/(intensity of reference beam of hologram H1). The “φ1” and “φ2” in FIGS. 17C and 17D are phases of object beams in holograms H1 and H2, respectively.
TABLE 1SequentialParallelDiversityschemeschemeschemeTime resolutionLowHighHighSpace resolutionHighLowHighDifference in intensity of objectLargeLargeSmallbeam between hologramsDifference in phase of object beamLargeLargeNegligiblebetween hologramsDifference in intensity of referenceNegligibleSmallSmallbeam between hologramsDifference in phase of referenceNegligibleNegligibleNegligiblebeam between holograms
As shown in Table 1, in the sequential scheme, the time resolution is low, and therefore, when the phase information of a moving object is measured, the intensity distribution and the phase distribution of the object beam may largely vary between holograms. Meanwhile, it can be said that, except for the phase difference to be given, only an intensity difference and a phase difference which are negligible in terms of time and space are caused since the reference beam is recorded without spatial division. Accordingly, in this case, measurement errors shown in FIGS. 17A to 17D may possibly be caused.
In the parallel scheme, since multiple holograms can be acquired by a single imaging, the time resolution is high. However, since a hologram is acquired in a spatially dividing manner in the parallel scheme, it is necessary to interpolate the intensity distribution and the phase distribution of an object beam and the intensity distribution of a reference beam, and thus an interpolation error is inevitably caused. Consequently, in the parallel scheme, a large measurement error may possibly be caused depending on the interpolation accuracy and the shape of the inspection object. In this case, the measurement errors shown in FIGS. 17A to 17F may possibly be caused.
The diversity scheme can achieve a high resolution, in terms of time and space. However, in the diversity scheme, a measurement error may be caused by a difference in intensity distributions of an object beam and a reference beam due to difference in sensitivity of imaging devices. In this case, the measurement errors shown in FIGS. 17A, 17B, 17E, and 17F may be caused.
As described, phase-shift digital holography has a potential to achieve a high-definition and highly accurate complex amplitude measurement, but at the same time has a problem that a measurement error is caused by non-uniformity in the intensity distributions of object beams and the reference beams between holograms. Therefore, a phase measurement method which can solve these problems is desired.